
If \(E\subset {\mathbb {R}}^n\)  is porous, then \(E\)  is weakly porous.

If \(E\subset {\mathbb {R}}^n\)  is weakly porous, then \(\vert E \vert = 0\) . This is a consequence
of the Lebesgue differentiation theorem.

\(E\subset {\mathbb {R}}^n\)  is weakly porous if and only if the closure \({\mathchoice{\unknown. {\displaystyle E}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle E}\displaystyle E}\hspace{0.0pt}}\) E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\( is weakly porous.\item Weak porosity implicitly implies that for every cube \) PRn\( there exists an \) E\(-free dyadic subcube \) QD(P)\(.\)
Let \(E\subset {\mathbb {R}}^n\)  be a nonempty set.
Given a cube \(P \subset {\mathbb {R}}^n\)  and \(\delta >0\) , we write
\(\widehat{\mathcal {F}_\delta }(P)= \lbrace  Q\in \mathcal {D}(P) \,:\, |Q| \ge \delta |\mathcal {M}(P)| \text{ and }Q \cap E = \emptyset \rbrace .\) 
We denote by \(\mathcal {F}_\delta (P)\)  the maximal subfamily of the cubes in \(\widehat{\mathcal {F}_\delta }(P).\) 
That is,
each \(R\in \widehat{\mathcal {F}_\delta }(P)\)  is
contained in some cube \(Q\in \mathcal {F}_\delta (P)\)  and
if \(Q\in \mathcal {F}_\delta (P)\) , then
\(Q\)  is not strictly contained in another cube in \(\widehat{\mathcal {F}_\delta }(P)\) .
Observe that the cubes in \(\mathcal {F}_\delta (P)\)  are pairwise disjoint, since two dyadic cubes are either
disjoint, or one of them is strictly contained in the other one.
The weak porosity of \(E\)  can now be formulated in terms of the sets \(\mathcal {F}_\delta \) , since \(E\)  is weakly porous if and only if
there are constants \(0<c,\delta <1\)  such that
\(\sum _{Q\in \mathcal {F}_\delta (P)} \vert Q\vert \ge c\vert P\vert \quad \text{ for all cubes } P\subset {\mathbb {R}}^n.\) 
Indeed, it is clear that
(REF ) implies weak porosity of \(E\) . Conversely,
\(c\vert P\vert \le \sum _{k=1}^N |Q_k|\le \sum _{Q\in \mathcal {\mathcal {F}_\delta }(P)} \sum _{k=1}^N\mathbf {1}_{Q_k\subset Q}\vert Q_k\vert \le \sum _{Q\in \mathcal {F}_\delta (P)} \vert Q\vert ,\) 

whenever \(c\) , \(\delta \) , \(P\)  and \(Q_k\) , \(k=1,\ldots ,N\) , are as in
Definition REF  (ii).
Part REF  of the
next lemma will be important when proving that weak porosity implies the \(A_1\) -property for
\(\operatorname{dist}(\cdot ,E)^{-\alpha }\) , for some \(\alpha >0\) ; see the proof of Lemma REF .
Lemma 3.2 
Assume that \(E\subset {\mathbb {R}}^n\)  is weakly porous set, with constants
\(0<c,\delta <1\) . Then the following
statements hold.


Assume that \(Q\subset R\)  are two
cubes such that \(E\cap Q\ne \emptyset \)  and
\(\vert \mathcal {M}(Q)\vert < 4^{-n}\delta \vert \mathcal {M}(R)\vert \) .
Then
\(\vert Q\vert \le (1-2^{-n}c)\vert R\vert .\) 


Assume
that \(Q\subset R\)  are two cubes such
that
\(\vert R\vert =2^n\vert Q\vert \) . Then there
exists a number \(k=k(n,c)\in {\mathbb {N}}\)  such that
\(\vert \mathcal {M}(R)\vert \le 4^{nk}\delta ^{-k}\vert \mathcal {M}(Q)\vert .\) 

 Assume that \(Q\subset R\)  are two cubes. Then there exist
constants \(C=C(n,c,\delta )\)  and \(\sigma =\sigma (n,c,\delta )>0\)  such that
\(\vert \mathcal {M}(R)\vert \le C\left( \frac{\ell (R)}{\ell (Q)} \right)^\sigma \vert \mathcal {M}(Q)\vert .\) 


We first remark that the dyadic grids \(\mathcal {D}(Q)\) 
and \(\mathcal {D}(R)\)  need not be compatible, and
this is taken into account in the arguments below.
First we show REF .
Fix \(S\in \mathcal {F}_\delta (R)\) . We claim that the
center \(x_S\in R\)  of \(S\)  belongs to \(R\setminus Q\) .
Assume the contrary, namely, that \(x_S\in Q\) .
Since \(S\)  is \(E\) -free and \(Q\)  intersects \(E\) , there
exists an \(E\) -free dyadic cube \(T\in \mathcal {D}(Q)\)  such
that \(\ell (T)\ge \ell (S)/4\) . It follows that
\(\vert \mathcal {M}(Q)\vert \ge \vert T\vert \ge 4^{-n}\vert S\vert \ge 4^{-n}\delta \vert \mathcal {M}(R)\vert .\) 
This is a contradiction, since
\(\vert \mathcal {M}(Q)\vert < 4^{-n}\delta \vert \mathcal {M}(R)\vert \)  by assumption.
We have shown that \(x_S\in R\setminus Q\) , and
therefore there
exists a cube \(S^{\prime }\subset S\setminus Q\)  such that
\(\vert S^{\prime }\vert = 2^{-n}\vert S\vert \) .
Since \(\lbrace S^{\prime }\,:\,S\in \mathcal {F}_\delta (R)\rbrace \)  is
a pairwise disjoint family of cubes contained in
\(R\setminus Q\) , we
obtain that
\(\vert R\vert - \vert Q\vert =\vert R\setminus Q\vert \ge \sum _{S\in \mathcal {F}_\delta (R)} \vert S^{\prime }\vert = 2^{-n} \sum _{S\in \mathcal {F}_\delta (R)} \vert S\vert .\) 

By weak porosity,
the last term above is bounded below by \(2^{-n}c\vert R\vert \) , and
reorganizing the terms gives
\((1-2^{-n}c)\vert R\vert \ge \vert Q\vert \)  as claimed
in REF .
Next we show REF .
If \(E\cap Q=\emptyset \) , then
\(\vert \mathcal {M}(R)\vert \le \vert R\vert =2^{n}\vert Q\vert \le 4^{n} \delta ^{-1}\vert Q\vert =4^n\delta ^{-1}\vert \mathcal {M}(Q)\vert .\) 
In this case, we may take \(k=1\) .
In the sequel we
assume that \(E\cap Q\ne \emptyset \) .
Choose \(k=k(n,c)\)  such that \(2^{n/k}<\frac{1}{1-2^{-n}c}\) .
Then there exists a finite sequence
\(Q=R_0\subset R_1\subset R_2\subset \cdots \subset R_k=R\) 
of cubes such that \(\vert R_i\vert \cdot \vert R_{i-1}\vert ^{-1} = 2^{n/k}\) .
Observe that
\(2^n=(2^{n/k})^k = \prod _{i=1}^k \frac{\vert R_i\vert }{\vert R_{i-1}\vert }=\frac{\vert R_k\vert }{\vert R_0\vert }=\frac{\vert R\vert }{\vert Q\vert }.\) 
Fix \(1\le i\le k\) .
We have \(\emptyset \ne E\cap Q\subset E\cap R_{i-1}\)  and \(R_{i-1}\subset R_i\) . Moreover,
\((1-2^{-n}c)\vert R_i\vert = (1-2^{-n}c) 2^{n/k}\vert R_{i-1}\vert < \vert R_{i-1}\vert \) 
and therefore the contrapositive of part REF  implies that
\(\vert \mathcal {M}(R_{i-1})\vert \ge 4^{-n}\delta \vert \mathcal {M}(R_i)\vert \) 
for all \(i=1,2,\ldots ,k\) . This allows us to conclude
that
\(\vert \mathcal {M}(R_0)\vert \ge 4^{-n}\delta \vert \mathcal {M}(R_1)\vert \ge (4^{-n}\delta )^2 \vert \mathcal {M}(R_2)\vert \ge \cdots \ge (4^{-n}\delta )^k\vert \mathcal {M}(R_k)\vert .\) 
The desired conclusion follows, since \(R_0=Q\)  and
\(R_k=R\) .
Finally, we prove REF . An easy computation shows that \(R \subset \lambda Q,\)  for \(\lambda = 3 \ell (R)/\ell (Q).\) 
Here \(\lambda Q\)  denotes the cube with the same center as \(Q\)  and side-length equal to \(\lambda \ell (Q).\) 
Then, for
\(m= 1+ \left\lfloor \log _2 \left( \frac{3\ell (R)}{\ell (Q)} \right) \right\rfloor ,\) 
we have that \(R \subset 2^m Q.\) 
Hence \(\vert \mathcal {M}(R)\vert \le C(n)\vert \mathcal {M}(2^mQ) \vert \) .
Denote by \(C_1=4^{nk}\delta ^{-k}\)  the constant in REF . Then, by iterating  REF  we obtain
\(\vert \mathcal {M}(2^mQ) \vert &\le C_1^m \vert \mathcal {M}(Q) \vert \le C_1^{1+\log _2 \left( \frac{3\ell (R)}{\ell (Q)} \right)}\vert \mathcal {M}(Q) \vert \\& = C(n,c,\delta )\left( \frac{\ell (R)}{\ell (Q)} \right)^\sigma \vert \mathcal {M}(Q)\vert ,\) 

where \(\sigma =\sigma (n,c,\delta )\) .
The claim REF  follows by combining the above estimates.
Example 3.3 Unlike for porous sets, inclusions do not preserve weak porosity: there are sets \(F \subset E\)  such that \(E\)  is weakly porous but \(F\)  is not.
For instance, \({\mathbb {Z}}\)  is clearly a weakly porous subset of \({\mathbb {R}}\) , but \({\mathbb {N}}\subset {\mathbb {Z}}\)  is not a weakly porous subset of \({\mathbb {R}}\) .
Indeed, assume for the contrary that \({\mathbb {N}}\)  is weakly porous in \({\mathbb {R}}\) 
with constants \(0<c,\delta <1\) .
Consider
cubes \(Q_j=[0,2^j)\) , \(j\in {\mathbb {N}}\) .
Observe that \(Q_j\subset R_j=[-2^j,2^j)\) .
Lemma REF  REF  implies
that there is a constant \(C=C(c,\delta )>0\)  such that
\(2^j=\vert \mathcal {M}(R_j)\vert \le C\vert \mathcal {M}(Q_j)\vert =C\) .
By choosing \(j\)  large enough, we get a contradiction.

\(A_1\)  implies weak porosity
This section and the following Section 
contain the proof of Theorem REF . We begin
by proving the necessity part of the equivalence
in the theorem, that is, if
\(\operatorname{dist}(\cdot ,E)^{-\alpha }\)  is an \(A_1\)  weight, then
\(E\)  is a weakly porous set. The straight-forward proof
illustrates in a nice way the connection between the \(A_1\)  condition and
the definition of weak porosity.
Lemma 4.1 
Let \(E\subset {\mathbb {R}}^n\)  be a nonempty set, let \(\alpha >0\) , and write
\(w(x)=\operatorname{dist}(x,E)^{-\alpha }\)  for all \(x\in {\mathbb {R}}^n\) . If
\(w\in A_1\) , then
\(E\)  is weakly porous with constants depending
on \(n\) , \(\alpha \)  and \([w]_{A_1}\) .

Since \(\operatorname{dist}(\cdot ,E)=\operatorname{dist}(\cdot ,{\mathchoice{\unknown. {\displaystyle E}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle E}\displaystyle E}\hspace{0.0pt}}\) E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E)\( and\) E\( is weakly porous if and only if \)\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\( is weakly porous, quantitatively, we may assume that \) E\( is closed.Assume that \) wA1\( and fix \) 0<<1\( to be chosen later.Let \) PRn\( be a cube andwrite \) =(M(P))\( for the sidelength of \) M(P)\(.\) Observe that the set \(E\)  is of measure zero, since \(w\)  is locally
integrable and \(w(x)=\infty \)  in \(E\) . Since \(E\)  is closed, for every \(x\in P\setminus E\)  we
have \(\operatorname{dist}(x,E)>0\)  and therefore there exists an \(E\) -free dyadic
cube \(Q\in \mathcal {D}(P)\)  such that \(x\in Q\) . As a consequence, we can write
\(P\setminus E\)  as a disjoint union of maximal \(E\) -free dyadic cubes \(Q\in \mathcal {D}(P)\) .
Let \(x\in P\setminus E\)  such that \(x\notin \bigcup _{Q\in \mathcal {F}_\delta (P)} Q\) . Then
the maximal \(E\) -free dyadic cube \(Q\in \mathcal {D}(P)\)  containing \(x\)  satisfies
\(\vert Q\vert <\delta \vert \mathcal {M}(P)\vert =\delta \ell ^n.\) 
Since \(\pi Q \in \mathcal {D}(P)\)  is not \(E\) -free, we have
\(\operatorname{dist}(x,E)\le \operatorname{diam}(\pi Q)< \delta ^{1/n}2\sqrt{n} \ell .\) 
It follows
that
\(\ell ^{-\alpha } < C(n,\alpha )\delta ^{\alpha /n}\operatorname{dist}(x,E)^{-\alpha }\) 
for every \(x\in (P\setminus E)\setminus \bigcup _{Q\in \mathcal {F}_\delta (P)} Q\) .
By integrating, and using the fact that \(E\)  is of measure zero, we obtain
\(\ell ^{-\alpha }\frac{\vert P\setminus \bigcup _{Q\in \mathcal {F}_\delta (P)} Q\vert }{\vert P\vert }&\le C(n,\alpha )\delta ^{\alpha /n}\frac{1}{\vert P\vert }\int _{P\setminus \bigcup _{Q\in \mathcal {F}_\delta (P)} Q}\operatorname{dist}(x,E)^{-\alpha }\,dx\\&\le C(n,\alpha )\delta ^{\alpha /n}\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}P}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P}}\operatorname{dist}(x,E)^{-\alpha }\,dx\\&\le C(n,\alpha )\delta ^{\alpha /n}[w]_{A_1}\operatornamewithlimits{ess\, inf}_{x\in P} \operatorname{dist}(x,E)^{-\alpha }.\) 

Denote by \(y\)  the center of \(\mathcal {M}(P)\subset P\) . Then
\(\operatornamewithlimits{ess\, inf}_{x\in P} \operatorname{dist}(x,E)^{-\alpha }\le \operatorname{dist}(y,E)^{-\alpha }\le 2^\alpha \ell \mathcal {M}(P))^{-\alpha }=2^\alpha \ell ^{-\alpha }.\) 
Simplifying, we get
\(\vert P\vert -\sum _{Q\in \mathcal {F}_\delta (P)} \vert Q\vert =\biggl \vert P\setminus \bigcup _{Q\in \mathcal {F}_\delta (P)} Q\biggr \vert \le C(n,\alpha )\delta ^{\alpha /n}[w]_{A_1}\vert P\vert .\) 
It remains to choose \(\delta =\delta (n,\alpha ,[w]_{A_1})>0\)  so small that \(C(n,\alpha )\delta ^{\alpha /n}[w]_{A_1}<1\) ,
and condition (REF ) follows.

Weak porosity implies \(A_1\)
Next, we turn to the sufficiency part of the equivalence
in Theorem REF , that is, the weak porosity of \(E\) 
implies that \(\operatorname{dist}(\cdot ,E)^{-\alpha }\)  is an \(A_1\)  weight;
see Lemma REF . The proof applies an iteration scheme,
which is built on an efficient use
of the dyadic definition of weak porosity;
see the proof of Lemma REF . The following sets \(\mathcal {F}_\delta ^k\)  and
\(\mathcal {G}_\delta ^k\)  will be important in the iteration.
Fix a weakly porous closed set \(E\subset {\mathbb {R}}^n\)  with constants \(0<c,\delta <1\)  and a cube \(P_0\subset {\mathbb {R}}^n\) .
Recall that \(\mathcal {F}_\delta (P_0)\)  is the maximal subfamily of the collection
\(\widehat{\mathcal {F}_\delta }(P_0)= \bigl \lbrace  Q\in \mathcal {D}(P_0) \,:\, |Q| \ge \delta |\mathcal {M}(P_0)| \text{ and } Q \cap E = \emptyset \bigr \rbrace .\) 
We will need also the complementary family \(\mathcal {G}_\delta (P_0)\) , which is
defined to be the maximal subfamily of the collection
\(\widehat{\mathcal {G}_\delta }(P_0)=\Biggl \lbrace P\in \mathcal {D}(P_0) \,:\,P\subset P_0\setminus \bigcup _{Q\in \mathcal {F}_\delta (P_0)}Q\Biggr \rbrace .\) 
Notice that
\(\vert Q\vert \ge \delta |\mathcal {M}(P_0)|\)  for all \(Q\in \mathcal {G}_\delta (P_0)\) .
We let
\(\mathcal {G}_\delta ^0=\lbrace P_0\rbrace \) , \(\mathcal {F}_\delta ^1=\mathcal {F}_\delta (P_0)\) , \(\mathcal {G}_\delta ^1=\mathcal {G}_\delta (P_0)\) ,
\(\mathcal {F}_\delta ^2=\bigcup _{R\in \mathcal {G}_\delta ^1} \mathcal {F}_\delta (R),\,\qquad \mathcal {G}_\delta ^2=\bigcup _{R\in \mathcal {G}_\delta ^1}\mathcal {G}_\delta (R),\) 
and in general, for \(k=3,4,\ldots \) , we define
\(\mathcal {F}_\delta ^k=\bigcup _{R\in \mathcal {G}_\delta ^{k-1}} \mathcal {F}_\delta (R),\,\qquad \mathcal {G}_\delta ^k=\bigcup _{R\in \mathcal {G}_\delta ^{k-1}}\mathcal {G}_\delta (R).\) 
Lemma 5.1 
Assume that \(E\subset {\mathbb {R}}^n\)  is a weakly porous
closed set with constants \(0<c,\delta <1\) .
Let \(P_0\subset {\mathbb {R}}^n\)  be a cube, and let sets
\(\mathcal {F}_\delta ^k\) , for \(k=1,2,\ldots \) , be as above.
Then
\(P_0\setminus E = \bigcup _{k=1}^{\infty }\bigcup _{Q\in \mathcal {F}_\delta ^k} Q.\) 

Let \(x\in P_0 \setminus E\) . Because \(E\)  is closed, there exists a dyadic cube \(Q \in \mathcal {D}(P_0)\)  such that \(x\in Q\)  and \(Q \cap E = \emptyset \) . We claim that \(Q \subset \bigcup _{k=1}^\infty \bigcup \mathcal {F}_\delta ^k\) .
Suppose, for the sake of contradiction, that \(Q\)  is not a subset of this union.
Because \(Q\lnot \subset \bigcup \mathcal {F}_\delta ^1\) , there exists \(R_1 \in \mathcal {G}_\delta ^1\)  containing \(Q\) .
Now \(Q\lnot \subset \bigcup \mathcal {F}_\delta (R_1)\) , as \(Q\lnot \subset \bigcup \mathcal {F}_\delta ^2\) .
Thus there exists \(R_2\in \mathcal {G}_\delta (R_1)\)  containing \(Q,\)  and again, \(Q\lnot \subset \bigcup \mathcal {F}_\delta (R_2)\) .
Repeating this argument, for every \(k\)  we obtain cubes
\(R_1 \supset R_2 \supset \cdots \supset R_k \supset Q\) 
with \(R_j \in \mathcal {G}_\delta (R_{j-1})\)  and such that \(Q\lnot \subset \bigcup \mathcal {F}_\delta (R_k)\) .
Also, because each \(R_j\)  is strictly
contained in \(R_{j-1},\)  we must have \(\vert R_j \vert \le 2^{-n} \vert R_{j-1} \vert .\)  Then \(Q\)  satisfies
\(\vert Q \vert < \delta \vert \mathcal {M}(R_k) \vert \le \delta \vert R_k \vert \le \frac{\delta }{2^{n(k-1)}} \vert R_1 \vert \le \frac{\delta }{2^{ nk}} \vert P_0 \vert .\) 
Letting \(k \rightarrow \infty ,\)  we derive a contradiction.
Lemma 5.2 
Assume that \(E\subset {\mathbb {R}}^n\)  is a weakly porous
closed set with constants \(0<c,\delta <1\) .
Let \(P_0\subset {\mathbb {R}}^n\)  be a cube and let sets \(\mathcal {F}_\delta ^k\) , for \(k=1,2,\ldots \) , be as above.
Then there are constants \(0<\gamma =\gamma (c,\delta ,n)<\frac{1}{n}\)  and \(C=C(c,\delta ,n)>0\)  such that
\(\sum _{k=1}^\infty \sum _{Q\in \mathcal {F}_\delta ^k}\vert Q\vert ^{1-\gamma }\le C\vert P_0\vert \vert \mathcal {M}(P_0)\vert ^{-\gamma }.\) 

Let \(0<\gamma <\frac{1}{n}\) , whose exact value will be fixed later;
we remark that both inequalities \(\gamma >0\)  and \(\gamma <\frac{1}{n}\) 
are be needed in Lemma REF  below.
By the definition of \(\mathcal {F}^k_\delta \) , we obtain
\(\begin{split}\sum _{Q\in \mathcal {F}_\delta ^k}\vert Q\vert ^{1-\gamma }&\le \sum _{R\in \mathcal {G}_\delta ^{k-1}}\sum _{Q\in \mathcal {F}_\delta (R)}\delta ^{-\gamma } \vert \mathcal {M}(R)\vert ^{-\gamma }\vert Q\vert \\&\le \delta ^{-\gamma } \sum _{R\in \mathcal {G}_\delta ^{k-1}} \vert \mathcal {M}(R)\vert ^{-\gamma }\vert R\vert ,\end{split}\) 
for every \(k=1,2,\ldots \) .
Next, we show by induction that
\(\sum _{R\in \mathcal {G}_\delta ^{k-1}}\vert \mathcal {M}(R)\vert ^{-\gamma }\vert R\vert \le ((1-c)(\sigma \delta )^{-\gamma })^{k-1}\vert \mathcal {M}(P_0)\vert ^{-\gamma }\vert P_0\vert \) 
for every \(k\in {\mathbb {N}}\) .
If \(k=1\) , this is immediate since \(\mathcal {G}_\delta ^{k-1}=\lbrace P_0\rbrace \) .
Then we assume that (REF ) holds for some \(k\in {\mathbb {N}}\) .
Fix \(R\in \mathcal {G}_\delta ^{k-1}\)  and let \(P\in \mathcal {G}_\delta (R)\) .
Since
\(P\)  is a maximal dyadic cube in \(R\setminus \bigcup _{Q\in \mathcal {F}_\delta (R)}Q\)  and \(\mathcal {F}_\delta (R)\ne \emptyset \)  by weak porosity,
the dyadic parent \(\pi P\)  intersects a cube \(Q\)  in \(\mathcal {F}_\delta (R)\) .
Since \(\pi P,Q\in \mathcal {D}(R)\) , we have \(\pi P\subset Q\)  or \(Q\subset \pi P\) 
by the nestedness
property (D3) of dyadic cubes.
Clearly \(\pi P\subset Q\)  is not possible, as this would lead to the contradiction
\(P\subset \pi P\subset Q\subset \bigcup _{Q^{\prime }\in \mathcal {F}_\delta (R)}Q^{\prime }\) .
Therefore \(Q\subset \pi P\) .
By Lemma REF  REF , there exists a constant
\(\sigma =\sigma (c,\delta ,n)>0\)  such that
\(\vert \mathcal {M}(P)\vert \ge \sigma \vert \mathcal {M}(\pi P)\vert .\) 
Using also the definition of \(\mathcal {F}_\delta (R)\) , we get
\(\vert \mathcal {M}(P)\vert \ge \sigma \vert \mathcal {M}(\pi P)\vert \ge \sigma \vert Q\vert \ge \sigma \delta \vert \mathcal {M}(R)\vert .\) 
On the other hand,
since \(E\)  is weakly porous, we have by (REF ) that
\(\sum _{P\in \mathcal {G}_\delta (R)}\vert P\vert =\biggl (\vert R\vert -\sum _{Q\in \mathcal {F}_\delta (R)} \vert Q\vert \biggr )\le (1-c)\vert R\vert .\) 
Applying the two estimates above
and the induction hypothesis (REF ) for \(k\) , we obtain
\(\sum _{P\in \mathcal {G}_\delta ^{k}}\vert \mathcal {M}(P)\vert ^{-\gamma }\vert P\vert &\le \sum _{R\in \mathcal {G}_\delta ^{k-1}} \sum _{P\in \mathcal {G}_\delta (R)}(\sigma \delta )^{-\gamma }\vert \mathcal {M}(R)\vert ^{-\gamma }\vert P\vert \\&\le (\sigma \delta )^{-\gamma }\sum _{R\in \mathcal {G}_\delta ^{k-1}} \vert \mathcal {M}(R)\vert ^{-\gamma }\sum _{P\in \mathcal {G}_\delta (R)}\vert P\vert \\&\le (\sigma \delta )^{-\gamma }\sum _{R\in \mathcal {G}_\delta ^{k-1}} \vert \mathcal {M}(R)\vert ^{-\gamma }(1-c)\vert R\vert \\&\le (1-c)(\sigma \delta )^{-\gamma }((1-c)(\sigma \delta )^{-\gamma })^{k-1}\vert \mathcal {M}(P_0)\vert ^{-\gamma }\vert P_0\vert \\&\le ((1-c)(\sigma \delta )^{-\gamma })^{k}\vert \mathcal {M}(P_0)\vert ^{-\gamma }\vert P_0\vert .\) 

This proves (REF ) for \(k+1\) ,
and thus the claim holds for every \(k\in {\mathbb {N}}\) ,
by the principle of induction.
Now choose
\(\gamma =\gamma (c,\delta ,n)\in (0,1/n)\)  to be such that
\((1-c)(\sigma \delta )^{-\gamma } < 1\) .
Observe that
\(\sum _{k=1}^\infty ((1-c)(\sigma \delta )^{-\gamma })^{k-1}= C(c,\sigma ,\delta ,\gamma ) = C(c,\delta ,n)< \infty .\) 
Hence, by using also (REF ) and (REF ),
we have
\(\sum _{k=1}^\infty \sum _{Q\in \mathcal {F}_\delta ^k}\vert Q\vert ^{1-\gamma }&\le \sum _{k=1}^\infty \delta ^{-\gamma } \sum _{R\in \mathcal {G}_\delta ^{k-1}} \vert \mathcal {M}(R)\vert ^{-\gamma }\vert R\vert \\&\le \sum _{k=1}^\infty \delta ^{-\gamma }((1-c)(\sigma \delta )^{-\gamma })^{k-1}\vert \mathcal {M}(P_0)\vert ^{-\gamma }\vert P_0\vert \\&\le \delta ^{-\gamma } \vert \mathcal {M}(P_0)\vert ^{-\gamma }\vert P_0\vert \sum _{k=1}^\infty ((1-c)(\sigma \delta )^{-\gamma })^{k-1}\\&\le C(c,\delta ,n) \vert P_0\vert \vert \mathcal {M}(P_0)\vert ^{-\gamma }. \) 
Lemma 5.3 
Assume that \(E\subset {\mathbb {R}}^n\)  is a weakly porous
set with constants \(0<c,\delta <1\) .
Then there are constants \(0<\alpha =\alpha (c,\delta ,n)<1\)  and \(C=C(n,c,\delta )\) 
such that \(\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_1({\mathbb {R}}^n)\) 
and \([\operatorname{dist}(\cdot ,E)^{-\alpha }]_{A_1}\le C\) .

Observe that
the closure \({\mathchoice{\unknown. {\displaystyle E}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle E}\displaystyle E}\hspace{0.0pt}}\) E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(is also weakly porous.Since \) dist(,E)=dist(,\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E)\(, we mayassume in the sequel that \) E\( is a weakly porous closed set.Throughout this proof \) C\( denotesa constant that can depend on \) n\(, \) c\( and \)\(.Let \) 0<=(n,c,)<1n\( be as in Lemma \ref {mainlemmasufficiency}.Fix a cube \) P0 Rn,\( and assume first that \) P0\( is not an \) E\(-free cube.Let sets \) Fk\(, for \) P0\( and \) k=1,2,...\(, be definedas above.\) Since \(\gamma n < 1\) , we have
for every \(E\) -free cube \(Q\)  the estimate
\(\int _Q \operatorname{dist}(x, E)^{-\gamma n} \, dx \le \int _Q \operatorname{dist}(x, \partial Q)^{-\gamma n} \, dx= C(\gamma ,n) \ell (Q)^{n-\gamma n} = C |Q|^{1-\gamma }.\) 
In particular, the upper bound
\(\gamma n < 1\)  implies that the second integral in (REF ) is finite.
Bearing in mind that \(\vert E \vert =0,\)  using
Lemma REF  and combining (REF ) with Lemma REF ,
we obtain
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}} \operatorname{dist}(x, E)^{-\gamma n} \, dx & =\frac{1}{\vert P_0 \vert } \int _{P_0\setminus E} \operatorname{dist}(x, E)^{-\gamma n} \, dx= \frac{1}{\vert P_0 \vert } \sum _{k=1}^\infty \sum _{Q\in \mathcal {F}_\delta ^k} \int _{Q} \operatorname{dist}(x, E)^{-\gamma n} \, dx \\& \le \frac{C}{\vert P_0 \vert } \sum _{k=1}^\infty \sum _{Q\in \mathcal {F}_\delta ^k} \vert Q\vert ^{1-\gamma } \le C \vert \mathcal {M}(P_0)\vert ^{-\gamma }.\) 

Let \(x\in P_0\setminus E\) .
Since \(E\)  is closed, the point \(x\)  is contained in
a maximal \(E\) -free
dyadic cube \(Q\in \mathcal {D}(P_0)\) .
Recall that \(P_0\)  is not
\(E\) -free, and so \(Q\)  is a strict
subcube of \(P_0\) . Furthermore \(\pi Q\)  is not \(E\) -free
due to maximality of \(Q\) .
This implies that
\(\operatorname{dist}(x,E) \le \operatorname{diam}(\pi Q) = 2\operatorname{diam}(Q)=2\sqrt{n}\, \ell (Q)\le 2\sqrt{n}\, \ell (\mathcal {M}(P_0)).\) 
Hence,
\(\operatornamewithlimits{ess\, inf}_{x\in {P_0}} \operatorname{dist}(x,E)^{-\gamma n} \ge (2 \sqrt{n})^{-\gamma n}\ell (\mathcal {M}(P_0))^{-\gamma n} = C(n,c,\delta ) \vert \mathcal {M}(P_0) \vert ^{-\gamma },\) 
and we conclude that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}} \operatorname{dist}(x, E)^{-\gamma n} \, dx \le C \, \operatornamewithlimits{ess\, inf}_{x\in {P_0}} \operatorname{dist}(x,E)^{-\gamma n}.\) 
It remains to consider the case where \(P_0\)  is an \(E\) -free cube. We study two situations separately.
If \(\operatorname{dist}(P_0,E)<2\operatorname{diam}(P_0),\)  then we have \(\operatorname{dist}(x,E) \le 3 \operatorname{diam}(P_0)\)  for every \(x\in P_0,\)  and so
\(\operatornamewithlimits{ess\, inf}_{x\in {P_0}} \operatorname{dist}(x,E)^{-\gamma n} \ge \left( 3 \operatorname{diam}(P_0) \right)^{-\gamma n} \ge C |P_0|^{-\gamma }.\) 
Using (REF ), together with this observation, we obtain
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}} \operatorname{dist}(x, E)^{-\gamma n} \, dx \le C |P_0|^{-\gamma } \le C \, \operatornamewithlimits{ess\, inf}_{x\in {P_0}} \operatorname{dist}(x,E)^{-\gamma n}.\) 
Finally, we consider
the case \(\operatorname{dist}(P_0,E)\ge 2\operatorname{diam}(P_0)\) . If \(x,y\in P_0\) , then
\(\operatorname{dist}(x,E) & \ge \operatorname{dist}(y,E)- \vert x-y\vert \ge \operatorname{dist}(y,E)-\operatorname{diam}(P_0)\\& \ge \operatorname{dist}(y,E) - \tfrac{1}{2} \operatorname{dist}(P_0,E) \ge \tfrac{1}{2} \operatorname{dist}(y,E).\) 

Hence,
\(\operatorname{dist}(x,E)^{-\gamma n}\le C \operatornamewithlimits{ess\, inf}_{y\in P_0} \operatorname{dist}(y,E)^{-\gamma n}\) 
for all \(x\in P_0\) , and so
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{P_0}} \operatorname{dist}(x, E)^{-\gamma n} \, dx \le C\operatornamewithlimits{ess\, inf}_{y\in P_0} \operatorname{dist}(y,E)^{-\gamma n}.\) 
By combining
estimates (REF ), (REF ), and (REF ), we see that \(\operatorname{dist}(\cdot ,E)^{-\gamma n} \in A_1({\mathbb {R}}^n)\) ,
and this proves the theorem with \(\alpha =\gamma n\) .

Muckenhoupt exponent
In this section, we introduce the concept of Muckenhoupt exponent
and explore its connections to weak porosity and the \(A_p\)  properties of distance weights,
for \(1\le p<\infty \) .
In particular, we prove our second main result, Theorem REF ,
at the end of this section.
For a bounded set \(A\subset {\mathbb {R}}^n\)  and \(r>0\) , we let \(N(A,r)\)  denote the minimal
number of open balls of radius \(r\)  that are needed to cover the set \(A\) .
Recall that the Assouad dimension \(\operatorname{dim_A}(E)\)  of \(E\subset {\mathbb {R}}^n\) 
is then the infimum of \(\lambda \ge 0\)  such that
\(N\bigl (E\cap B(x,R),r\bigr )\le C\biggl (\frac{R}{r}\biggr )^{\lambda }\) 
for every \(x\in E\)  and \(0<r<R\) .
Equivalently, \(\operatorname{dim_A}(E)=n-\operatorname{{codim}_A}(E)\) , where
the Assouad codimension
\(\operatorname{{codim}_A}(E)\)  is the supremum of \(\alpha \ge 0\)  such that
\(\frac{\vert E_r\cap B(x,R)\vert }{\vert B(x,R)\vert } \le C\biggl (\frac{R}{r}\biggr )^{-\alpha }\) 
for every \(x\in E\)  and \(0<r<R\) .
Here
\(E_r=\lbrace y\in {\mathbb {R}}^n:\operatorname{dist}(y,E)<r\rbrace \) 
is the open \(r\) -neighborhood of \(E\).
See e.g. [1]}
for more details concerning this equivalence, which
also follows from Lemma REF .
It is well-known that a set \(E\subset {\mathbb {R}}^n\)  is porous if and only if
\(\operatorname{dim_A}(E)<n\) , or equivalently \(\operatorname{{codim}_A}(E)>0\) ,
as was already pointed out in the introduction.
See e.g. [2]}
or [3]}
for details.
The following Muckenhoupt exponent
can be seen as a refinement of the Assouad codimension:
for porous sets these two agree
but the Muckenhoupt exponent can be nonzero also for nonporous sets;
see the comment after Definition REF .
Definition 6.1 
Let \(E\subset {\mathbb {R}}^n\) .
(i)
If \(B(x,r)\)  is a ball in \({\mathbb {R}}^n\) , we denote
by \(h_E(B(x,r))\)  the supremum of all \(t>0\) 
such that \(B(y,t)\subset B(x,r)\setminus E\)  for some \(y\in B(x,r)\) .
If there is no such number \(t>0\) , then
we set \(h_E(B(x,r))=0\) .
(ii)
If \(h_E(B(x,R))>0\)  for every \(x\in E\)  and \(R>0\) ,
then the
Muckenhoupt exponent \(\operatorname{{Mu}}(E)\) 
is the supremum of the numbers \(\alpha \in {\mathbb {R}}\) 
for which there exists a constant \(C\)  such that
\(\frac{\vert E_r\cap B(x,R)\vert }{\vert B(x,R)\vert } \le C\biggl (\frac{h_E(B(x,R))}{r}\biggr )^{-\alpha }\) 
for every \(x\in E\)  and \(0<r < h_E(B(x,R)) \le R\) .
If \(h_E(B(x,R))=0\)  for some \(x\in E\)  and \(R>0\) ,
then we set \(\operatorname{{Mu}}(E)=0\) .


Observe that
\(h_E(B(x,R))\le R/2\)  if \(x\in E\) .
It is clear from the definition that \(\operatorname{{Mu}}(E)\ge 0\)  for all sets \(E\subset {\mathbb {R}}^n\) ,
since (REF ) always holds with \(\alpha =0\)  if \(h_E(B(x,R))>0\) .
If \(E\subset {\mathbb {R}}^n\)  is porous, then
\(cR\le h_E(B(x,R))\le R/2\)  for all \(x\in E\)  and \(R>0\) ,
showing that \(\operatorname{{Mu}}(E)=\operatorname{{codim}_A}(E)\) .
On the other hand, if \(E\subset {\mathbb {R}}^n\)  is not porous,
then \(\operatorname{{codim}_A}(E)=0\le \operatorname{{Mu}}(E)\) , and thus always \(\operatorname{{codim}_A}(E)\le \operatorname{{Mu}}(E)\) .
This inequality is strict if and only if \(E\)  is weakly porous but not porous
since the weak
porosity of \(E\)  is characterized by \(\operatorname{{Mu}}(E)>0\) ,
see Corollary REF .
As an example, it is straightforward to see that \(\operatorname{{codim}_A}({\mathbb {Z}})=0\)  and \(\operatorname{{Mu}}({\mathbb {Z}})=1\) .
See also Section  for other examples of such sets.
In Lemma REF  below we give for the Muckenhoupt exponent an alternative characterization,
which resembles the definition of the Assouad dimension. The following estimate will be applied in the
proof of Lemma REF .
Lemma 6.2 
Let \(E\subset {\mathbb {R}}^n\) , \(x\in E\)  and \(0<r<R\) . Then
\(C_1(n) N\bigl (E\cap B(x,R/2),r\bigr )\le \frac{\vert E_r\cap B(x,R)\vert }{r^n}\le C_2(n) N\bigl (E\cap B(x,2R),r\bigr ).\) 

Let \(\lbrace B(x_i,r)\rbrace _{i=1}^N\)  be a cover of \(E\cap B(x,2R)\) , with
\(N=N\bigl (E\cap B(x,2R),r\bigr )\) . Then
\(E_r\cap B(x,R)\subset \bigcup _{i=1}^N B(x_i,2r),\) 
and thus
\(\vert E_r\cap B(x,R)\vert \le C(n) N(2r)^n= C_2(n)r^n N\bigl (E\cap B(x,2R),r\bigr ).\) 
This proves the second inequality in the claim.
Conversely, let
\(\lbrace B(x_i,r)\rbrace _{i=1}^N\)  be a cover of \(E\cap B(x,R/2)\) 
such that \(x_i\in E\cap B(x,R/2)\)  for all \(i=1,\ldots ,N\)  and the balls
\(B(x_i,r/2)\)  are pairwise disjoint (such a cover can be found by
choosing \(\lbrace x_i\rbrace _{i=1}^N\)  to be a maximal \(r\) -net in
\(E\cap B(x,R/2)\) , see [4]}).
Then
\(E_r\cap B(x,R)\supset \bigcup _{i=1}^N B(x_i,r/2),\) 
and thus
\(\vert E_r\cap B(x,R)\vert \ge C(n) N(r/2)^n\ge C_1(n) r^n N\bigl (E\cap B(x,R/2),r\bigr ).\) 
This proves the first inequality in the claim.
Lemma 6.3 
Let \(E\subset {\mathbb {R}}^n\)  be such
that \(h_E(B(x,R))>0\)  for every \(x\in E\)  and \(R>0\) .
Then \(\operatorname{{Mu}}(E)\) 
is the supremum of the numbers \(\alpha \ge 0\) 
for which there exists a constant \(C\)  such that
\(N\bigl (E\cap B(x,R),r\bigr )\le C\biggl (\frac{R}{r}\biggr )^{n}\biggl (\frac{h_E(B(x,R))}{r}\biggr )^{-\alpha }\) 
for every \(x\in E\)  and \(0<r < h_E(B(x,R)) \le R\) .

Assume first that \(\alpha \ge 0\)  is such that (REF )
holds for every \(x\in E\)  and \(0<r < h_E(B(x,R))\le R\)  with a constant \(C_1\) .
Let \(x\in E\)  and \(0<r < h_E(B(x,R))\le R\) .
Then
\(0<r < h_E(B(x,R))\le h_E(B(x,2R)) \le R<2R\) , and
by Lemma REF  and (REF ) we have
\(\frac{\vert E_r\cap B(x,R)\vert }{\vert B(x,R)\vert }& \le C(n) \biggl (\frac{r}{R}\biggr )^{n} N\bigl (E\cap B(x,2R),r\bigr )\\& \le C_1 C(n) \biggl (\frac{r}{R}\biggr )^{n} \biggl (\frac{2R}{r}\biggr )^{n}\biggl (\frac{h_E(B(x,2R))}{r}\biggr )^{-\alpha }\\& \le C(n,C_1) \biggl (\frac{h_E(B(x,R))}{r}\biggr )^{-\alpha }.\) 

Thus \(\alpha \le \operatorname{{Mu}}(E)\) .
Lemma REF  implies that
(REF ) holds with \(\alpha =0\)  and
hence \(\operatorname{{Mu}}(E)\ge 0\) .
If \(\operatorname{{Mu}}(E)=0\)  and (REF ) holds for \(\alpha \ge 0,\)  the preceding computation shows that \(\alpha =0\)  as well, and the result follows.
Then assume that \(0\le \alpha < \operatorname{{Mu}}(E)\) 
and let \(x\in E\)  and \(0<r < h_E(B(x,R))\le R\) . By Lemma REF  and (REF ),
for \(\alpha \)  and a constant \(C_\alpha \) ,
we have
\(N\bigl (E\cap B(x,R),r\bigr )& \le C(n)\frac{\vert E_r\cap B(x,2R)\vert }{r^n}\\& \le C(n)C_\alpha \biggl (\frac{2R}{r}\biggr )^{n}\biggl (\frac{h_E(B(x,2R))}{r}\biggr )^{-\alpha }\\& \le C(n,C_\alpha )\biggl (\frac{R}{r}\biggr )^{n}\biggl (\frac{h_E(B(x,R))}{r}\biggr )^{-\alpha }.\) 

Since this holds for every \(0\le \alpha < \operatorname{{Mu}}(E)\) , we conclude that
\(\operatorname{{Mu}}(E)\)  is indeed the supremum of \(\alpha \)  for which
(REF ) holds for all \(x\in E\)  and \(0<r < h_E(B(x,R))\le R\) .
Next, we turn to the relations between the Muckenhoupt exponent and \(A_1\)  weights.
Lemma REF  and Theorem REF  together characterize the property
\(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) , for \(\alpha >0\) , in terms of the Muckenhoupt exponent of \(E\) ; see
Theorem REF .
Lemma 6.4 
Let \(E\subset {\mathbb {R}}^n\)  be a nonempty set and let \(\alpha \in {\mathbb {R}}\)  be
such that \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) . Then \(0\le \alpha \le \operatorname{{Mu}}(E)\) .

Assume first that \(\alpha <0\) .
Let \(x\in E\)  and \(r>0\) .
Then
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q(x,r)}} \operatorname{dist}(y,E)^{-\alpha }\,dy \le C\, \operatornamewithlimits{ess\, inf}_{y\in Q(x,r)} \operatorname{dist}(y,E)^{-\alpha }=0;\) 
here the cube \(Q(x,r)\)  is as in (REF ).
Thus \(\operatorname{dist}(y,E)^{-\alpha }=0\)  for almost every \(y\in Q(x,r)\) ,
which is a contradiction since \(\operatorname{dist}(\cdot ,E)^{-\alpha }\)  is a weight.
Hence \(\alpha \ge 0\) .
The claim holds if \(\alpha =0\) , and so we may assume that \(\alpha >0\) .
Then \(h_E(B(x,R))>0\)  for every \(x\in E\)  and \(R>0\) .
Indeed, otherwise there exists a ball \(B(x,R)\)  such
that \(\operatorname{dist}(y,E)=0\)  for every \(y\in B(x,R)\) ,
and therefore \(\operatorname{dist}(\cdot ,E)^{-\alpha }\) 
is not locally integrable. This
is again a contradiction since \(\operatorname{dist}(\cdot ,E)^{-\alpha }\)  is a weight.
Let \(x\in E\)  and \(0<r < h_E(B(x,R))\le R\) , and write \(F= E_r \cap B(x,R)\) .
Let
\(C_1\)  be the constant in the \(A_1\)  condition (REF ) for \(\operatorname{dist}(\cdot ,E)^{-\alpha }\) .
Observe from \(B(x,R)\subset Q(x,R)\)  that
\(h_E(B(x,R))\le \operatornamewithlimits{ess\, sup}_{y\in Q(x,R)}\operatorname{dist}(y,E),\) 
and hence
\(\operatornamewithlimits{ess\, inf}_{y\in Q(x,R) }\operatorname{dist}(y,E)^{-\alpha }\le h_E(B(x,R))^{-\alpha }.\) 
Since \(\operatorname{dist}(y,E)< r\) 
for every \(y\in F\)  and
\(F\subset B(x,R) \subset Q(x,R),\) 
using the \(A_1\)  condition (REF )
we obtain
\(\begin{split}\vert F\vert & \le r^{\alpha } \int _{F} \operatorname{dist}(y,E)^{-\alpha }\,dy \le r^{\alpha } \int _{Q(x,R)} \operatorname{dist}(y,E)^{-\alpha }\,dy \\& \le C_1 r^{\alpha } \vert Q(x,R)\vert h_E(B(x,R))^{-\alpha }= C(n,C_1)R^{n}\biggl (\frac{h_E(B(x,R))}{r}\biggr )^{-\alpha }.\end{split}\) 
Thus
\(\frac{\vert E_r \cap B(x,R) \vert }{\vert B(x,R)\vert }=\frac{\vert F \vert }{\vert B(x,R)\vert } \le C(n,C_1)\biggl (\frac{h_E(B(x,R))}{r}\biggr )^{-\alpha },\) 
and the claim \(\operatorname{{Mu}}(E)\ge \alpha \)  follows.
Theorem 6.5 
Let \(E\subset {\mathbb {R}}^n\)  be a nonempty
set
and assume that \(0\le \alpha < \operatorname{{Mu}}(E)\) . Then
\(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) .

It suffices to
show that there
exists a constant \(C>0\)  such that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}} \operatorname{dist}(y,E)^{-\alpha }\,dy\le C \operatornamewithlimits{ess\, inf}_{y\in B(x,r)}\operatorname{dist}(y,E)^{-\alpha }\) 
for all
\(x\in E\)  and \(r>0\) .
Indeed, if \(\operatorname{dist}(Q,E)< 2\operatorname{diam}(Q)\)  for a cube \(Q\subset {\mathbb {R}}^n\) , then the
desired \(A_1\)  property (REF ) for \(w=\operatorname{dist}(\cdot ,E)^{-\alpha }\)  follows easily from (REF )
by considering a
ball \(B=B(x,r)\)  such that \(x\in E\) ,
\(Q\subset B\)  and \(\vert B\vert \le C(n)\vert Q\vert \) .
On the other hand, if \(\operatorname{dist}(Q,E)\ge 2\operatorname{diam}(Q)\) , then
an argument similar to the one leading to (REF ) shows that (REF ) holds,
and thus \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) .
Let \(\lambda >0\)  with \(\operatorname{{Mu}}(E)>\lambda >\alpha \) ,
and let \(x\in E\)  and \(r>0\) .
Observe from inequality \(\operatorname{{Mu}}(E)>0\)  that \(0<h_E(B(x,2r))\le r\) .
Hence, there is \(j_0\in {\mathbb {N}}\) 
such that
\(2^{-j_0} r < h_E(B(x,2r)) \le 2^{1-j_0} r.\) 
Define
\(F_j=\lbrace y\in B(x,r) : \operatorname{dist}(y,E) \le 2^{1-j}r \rbrace  \quad \text{ and }\quad A_{j}=F_{j}\setminus F_{j+1},\) 
for \(j\ge j_0\) .
Since \(\lambda <\operatorname{{Mu}}(E)\) , there is a constant \(C_1=C_1(E,\lambda ,n)\)  such that
\(\begin{split}\frac{\vert F_{j}\vert }{\vert B(x,r)\vert }&\le \frac{2^n\vert E_{2^{2-j}r}\cap B(x,2r)\vert }{\vert B(x,2r)\vert }\\&\le C_1 \biggl (\frac{h_E(B(x,2r))}{2^{-j} r}\biggr )^{-\lambda }= C_1 2^{-j\lambda }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }.\end{split}\) 
Since \(\lambda >0\)  and \({\mathchoice{\unknown. {\displaystyle E}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle E}\displaystyle E}\hspace{0.0pt}}\) E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)EB(x,r)Fj\( for every \) jj0\(,by letting \) j\( we see in particular that \)\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)EB(x,r) = 0\(.Here \) r>0\( is arbitrary, and thus \)\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E=0\(.\) If \(y\in B(x,r)\setminus {\mathchoice{\unknown. {\displaystyle E}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle E}\displaystyle E}\hspace{0.0pt}}\) E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(, then\) dist(y,E)y-x<r\(. Hence,\)\(B(y,\operatorname{dist}(y,E))\subset B(x,2r)\setminus E,\)\(and therefore \) 0<dist(y,E)hE(B(x,2r))21-j0r\(.It follows thatthe union of sets \) Aj\( with \) jj0\( cover \) B(x,r)\( up to the set\)\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)E\(\) E\(\) 

E
\(\)EB(x,r)\(, which has measure zero.If \) yAj\(, then\) 2-jr < dist(y,E) 21-jr\(.In addition, \) AjFj\( for every \) jj0\(.By combining the above observations and using~(\ref {e.level_est}) we obtain\begin{equation*}\begin{split}\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{B(x,r)}} \operatorname{dist}(y,E)^{-\alpha }\,dy&\le \frac{1}{\vert B(x,r)\vert } \sum _{j=j_0}^{\infty }\int _{A_{j}}\operatorname{dist}(y,E)^{-\alpha }\,dy \le \sum _{j=j_0}^{\infty }\frac{\vert F_{j}\vert }{\vert B(x,r)\vert }(2^{-j}r)^{-\alpha }\\&\le C_1 \sum _{j=j_0}^{\infty }(2^{-j}r)^{-\alpha } 2^{-j\lambda }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }\\&\le C_1 r^{-\alpha }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }\sum _{j=j_0}^{\infty }(2^{-j})^{\lambda -\alpha }\\&\le C(C_1,\lambda ,\alpha ) r^{-\alpha }\biggl (\frac{h_E(B(x,2r))}{r}\biggr )^{-\lambda }\left(\frac{h_E(B(x,2r))}{r}\right)^{\lambda -\alpha }\\&\le C(C_1,\lambda ,\alpha ) h_E(B(x,2r))^{-\alpha }\\&\le C(C_1,\lambda ,\alpha ) \operatornamewithlimits{ess\, inf}_{y\in B(x,r)}\operatorname{dist}(y,E)^{-\alpha }.\end{split}\end{equation*}This shows that (\ref {e.suff}) holds, and the claim follows.\)
Theorem 6.6 
Let \(\alpha >0\)  and let \(E\subset {\mathbb {R}}^n\)  be a nonempty
set. Then \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\)  if and only if \(\operatorname{{Mu}}(E)>\alpha \) .

If \(\operatorname{{Mu}}(E)>\alpha \) , then \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\)  by Theorem REF .
Conversely, assume that \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) .
Then, by the self-improvement of \(A_1\)  weights (see [5]}),
there exists \(s>1\)  such that \(\operatorname{dist}(\cdot ,E)^{-s\alpha }\in A_1\) .
Lemma REF  implies that \(\operatorname{{Mu}}(E)\ge s\alpha >\alpha \) .
Since \(\operatorname{dist}(\cdot ,E)^{0} \in A_1\)  holds for all (nonempty) sets \(E\subset {\mathbb {R}}^n\) 
(under the interpretation that \(0^0=1\) ), Theorem REF 
implies that
\(\operatorname{{Mu}}(E)=\sup \lbrace  \alpha \ge 0 : \operatorname{dist}(\cdot ,E)^{-\alpha } \in A_1 \rbrace \) 
for all nonempty sets \(E\subset {\mathbb {R}}^n\) . On the other hand,
by Theorem REF  we have \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) , for some \(\alpha >0\) , if and only if
\(E\)  is weakly porous. This, together with Theorem REF , gives the following corollary.
Corollary 6.7 
A nonempty set \(E\subset {\mathbb {R}}^n\)  is weakly porous
if and only if \(\operatorname{{Mu}}(E)>0\) .

We are now ready for the proof of our second main result.
Since \(E\)  is weakly porous,
we have \(\operatorname{{Mu}}(E)>0\)  by Corollary REF .
Therefore, the equivalences in both (i) and (ii) hold if \(\alpha =0,\) 
and so we may assume from now on that \(\alpha \ne 0.\)
If \(w =\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\) , then \(\alpha \ge 0\)  by Lemma REF .
Thus it suffices to consider the case \(\alpha >0\) , and then the claim in (i) follows
directly from Theorem REF .
In part (ii), let \(1<p<\infty \)  and
assume first that \(w\in A_p\) . Because \(E\)  is weakly porous, Lemma REF 
provides us with some \(\sigma >0\)  for which \(\operatorname{dist}(\cdot ,E)^{-\sigma } \in A_1({\mathbb {R}}^n)\) . If \(\alpha > 0,\)  we can use
Lemma REF  with \(\beta =\sigma /\alpha \)  to deduce that \(w=\operatorname{dist}(\cdot , E)^{-\alpha }\in A_1.\) 
Then Theorem REF  implies \(\operatorname{{Mu}}(E) >\alpha ,\)  and so (REF ) holds.
On the other hand, if \(\alpha <0,\)  then
we have
\(\operatorname{dist}(\cdot ,E)^{- \left(\frac{-\alpha }{p-1}\right) }= w^{1-p^{\prime }} \in A_{p^{\prime }},\) 
where \(\frac{-\alpha }{p-1} >0.\) 
Hence the previous case, for a positive power and the class \(A_{p^{\prime }}\) ,
shows that
\((1-p^{\prime })\operatorname{{Mu}}(E) < 0 < \frac{-\alpha }{p-1} < \operatorname{{Mu}}(E),\) 
which is equivalent to (REF ).
Conversely, assume that (REF ) holds for some \(\alpha \ne 0.\)  If \(\alpha > 0,\) 
then \(w=\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_1\subset A_p\)  by Theorem REF .
Finally, if \(\alpha <0,\)  we observe that (REF ) is equivalent to (REF ),
where \(\frac{-\alpha }{p-1} >0\) . Thus we may apply the preceding case for the exponent \(\frac{-\alpha }{p-1}>0\)  and the class \(A_{p^{\prime }}\)  to conclude that \(\operatorname{dist}(\cdot , E)^{\alpha /(p-1)} \in A_{p^{\prime }}.\)  Hence \(w = \operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p,\) 
proving part (ii).
Remark 6.8 
Note that in part (i) of Theorem REF  the explicit assumption that
\(E\)  is weakly porous is needed in the necessity part, since for \(\alpha =0\) 
the claim \(w\in A_1\)  holds for all (nonempty) sets \(E\subset {\mathbb {R}}^n\) . However, if \(\alpha >0\) ,
then we know by Theorem REF 
that \(w\in A_1\)  can only hold if \(E\)  is weakly porous,
which in turn is equivalent to \(\operatorname{{Mu}}(E)>0\) .
In part (ii) the case \(\alpha =0\)  again shows that (REF ) is
not necessary for \(w\in A_p\) , for general sets \(E\subset {\mathbb {R}}^n\) .
Moreover, if we do not assume weak porosity of \(E\) , then even
in the case \(\alpha \ne 0\)  the requirement (REF )
is not necessary for \(w\in A_p\) . This follows from Theorem REF ,
which gives
a set \(E\subset {\mathbb {R}}\)  with \(\operatorname{{Mu}}(E)=0\) ,
i.e. \(E\)  is not weakly porous, such that
\(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_p\)  for all \(0<\alpha <1\)  and all \(1<p<\infty \) .

Remark 6.9 
When \(E\subset {\mathbb {R}}^n\)  is a bounded set, the
upper Minkowski (or box) dimension
\(\operatorname{\overline{dim}_M}(E)\) 
is the infimum of all \(\lambda \ge 0\)  for which there is a constant \(C\)  such that
\(N(E,r)\le Cr^{-\lambda }\) 
for every \(0<r<\operatorname{diam}(E)\) .
Note that (REF )
is equivalent to the condition that there is a constant \(C\)  such that
\(\vert E_r\vert \le Cr^{n-\lambda }\) 
for every \(0<r<\operatorname{diam}(E)\) ; this follows from Lemma REF .
If a set \(E\subset {\mathbb {R}}^n\)  is weakly porous and \(0<\alpha <\operatorname{{Mu}}(E)\) ,
then \(\operatorname{dist}(\cdot ,E)^{-\alpha }\in A_1\)  by Theorem REF ,
and so \(\int _{B(x,R)} \operatorname{dist}(y,E)^{-\alpha }\,dy<\infty \) 
for every \(x\in E\)  and \(R>0\) . Hence, if \(x\in E\)  and \(R>0\) , then
it holds for all \(0<r<\operatorname{diam}(E\cap B(x,R))\le 2R\)  that
\(\vert (E\cap B(x,R))_r\vert \le r^{\alpha }\int _{B(x,3R)} \operatorname{dist}(y,E)^{-\alpha }\,dy\le C(x,R,E)r^{n-(n-\alpha )}.\) 

Thus
\(\operatorname{\overline{dim}_M}(E\cap B(x,R))\le n-\alpha <n.\) 
Since this holds for all \(0<\alpha <\operatorname{{Mu}}(E)\) ,
we obtain \(\operatorname{\overline{dim}_M}(E\cap B(x,R))\le n-\operatorname{{Mu}}(E)\) .
In particular, if \(E\subset {\mathbb {R}}^n\)  is bounded, then
\(0\le \operatorname{{Mu}}(E)\le n - \operatorname{\overline{dim}_M}(E)\) .
On the other hand, the condition that \(\operatorname{\overline{dim}_M}(E\cap B(x,R))\le c < n\)  for every \(x\in E\)  and \(R>0\)  is not sufficient
for the weak porosity of \(E\) . For instance, if \(E\subset {\mathbb {Z}}\subset {\mathbb {R}}\)  is not weakly porous
(e.g. \(E={\mathbb {N}}\) ), then we have \(\operatorname{\overline{dim}_M}(E\cap B(x,R)) = 0 < 1 = n\) 
for every \(x\in E\)  and \(R>0\)  since \(E\cap B(x,R)\)  is a finite set.
See also [6]} and the references therein for much more elaborate
connections between Minkowski dimensions and the integrability of distance functions.

Example of a weakly porous set
The notions of weak porosity and Muckenhoupt exponent are interesting only if there are
(plenty of) weakly porous sets which are not porous. Below we construct a family of such sets in \({\mathbb {R}}^n\) 
and determine the Muckenhoupt exponents for different values of the parameter \(\gamma >0\) .
These sets are inspired by the often used one-dimensional
example \(\lbrace j^{-\gamma }:j\in {\mathbb {N}}\rbrace \cup \lbrace 0 \rbrace \subset {\mathbb {R}}\) .
Theorem 7.1 
Let \(n\in {\mathbb {N}}\)  and \(\gamma >0\) . Then
the set
\(E=\bigcup _{j = 1} ^\infty \partial B\big (0, j^{-\gamma } \big ) \cup \lbrace 0 \rbrace \subset {\mathbb {R}}^n\) 
is weakly porous with \(\operatorname{{Mu}}(E)=\min \lbrace 1, \frac{n \gamma }{1+\gamma } \rbrace .\)

The origin is included in \(E\)  in order to have a compact set,
but for our purposes this does not make any essential difference.
See Figure REF  for an illustration of
the set
\(E\) .
<FIGURE>By considering the balls \(B(0,j^{-\gamma })\)  as \(j\rightarrow \infty \) , it is
straightforward to verify that \(E\)  is not porous,
and hence \(\operatorname{dim_A}(E)=n\) . Moreover,
special cases of the
computations in the proof of
Theorem REF  below
can be used to show that \(\operatorname{\overline{dim}_M}(E)=\max \lbrace n-1,\frac{n}{1+\gamma }\rbrace \) ,
and so in combination with Theorem REF 
we obtain for the set \(E\)  the identity \(\operatorname{{Mu}}(E) = n - \operatorname{\overline{dim}_M}(E)\) ;
compare to Remark REF .
For the proof of Theorem REF , we
define \(S^t=\partial B(0,t)\)  and \(A_s^t={\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B ( 0, t ) B(0, s )\( for every \) 0s t\(, where we use the notation \) B(0,0)=\(.We begin with the following lemma.\)
Lemma 7.2 
Let \(B=B(x,R) \subset {\mathbb {R}}^n\)  be a ball such that \(x \in S^t,\)  with \(t=j^{-\gamma }\)  for some \(j\in {\mathbb {N}}\) , and \(B \cap E = B \cap S^t.\)  Then (REF ) holds for \(B\)  if and only if \(\alpha \le 1\) . Moreover, if \(\alpha \le 1\) , then
the constant in (REF ) for \(B\) 
can be chosen to depend on \(n\) , \(\gamma \)  and \(\alpha \)  only.

We have \(h_E(B) = R/2,\)  and given \(0<r<h_E(B),\)  the set \(A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B\)  satisfies
\((2r) \inf _{b\in [t-r,t+r]} \mathcal {H}^{n-1}\left( S^{b} \cap B \right) \le \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \le (2r) \sup _{b\in [t-r,t+r]} \mathcal {H}^{n-1}\left( S^{b} \cap B \right),\) 
where \(\mathcal {H}^{n-1}\)  is the normalized
Hausdorff measure in \({\mathbb {R}}^n\) .
For each \(b\in [t-r,t+r],\)  the set \(S^b \cap B\)  is a hyperspherical cap within the sphere \(S^{b},\)  whose angle \(\alpha _{b}\)  satisfies, by virtue of the law of cosines, that \(\cos (\alpha _b) = \frac{b^2+t^2-R^2}{2bt}.\)  Therefore
\(\sin \left( \frac{\alpha _b}{2} \right) = \left( \frac{R^2-(b-t)^2}{4bt} \right)^{1/2}.\) 
For a sufficiently small constant \(c(\gamma ),\)  we have that \(r \le c(\gamma ) h_E(B)\)  implies \(\alpha _b \simeq C(\gamma ) \left( \frac{R }{2t} \right)\)  for every \(b\in [t-r,t+r]\) ;
here and below \(a\simeq C(*) b\)  means that \(C(*)^{-1}b\le a\le C(*)b\) . This leads us to
\(\mathcal {H}^{n-1}\big ( S^{b} \cap B \big ) \simeq C(n,\gamma ) b^{n-1} \left( \alpha _b \right)^{n-1} \simeq C(n,\gamma ) b^{n-1} \left( \frac{R}{2t} \right)^{ n-1 } \simeq C(n,\gamma ) R^{n-1},\) 
for every \(b\in [t-r,t+r].\)  The sets \(A_{(j-1)^{-\gamma }-r}^{(j-1)^{-\gamma }} \cap B\)  and \(A_{(j+1)^{-\gamma }}^{(j+1)^{-\gamma }+r} \cap B\)  (meaning \(A_{(j-1)^{-\gamma }-r}^{(j-1)^{-\gamma }} = \emptyset \)  in the case \(j=1\) ) are also contained in \(E_r \cap B\) , but their measures are controlled by \(C(n,\gamma ) \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert .\)  Bearing in mind this observation and (REF ) and (REF ), we obtain
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } \le C(n,\gamma ,\alpha ) R^{\alpha -n} r^{-\alpha } \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \le C(n,\gamma ,\alpha ) \left( \frac{r}{R} \right)^{1-\alpha }.\) 
If \(\alpha \le 1,\)  the last term is bounded by \(C(n,\gamma ,\alpha ).\)  On the other hand, if \(\alpha > 1,\)  then (REF ) and (REF ) yield
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } \ge c(n,\gamma ,\alpha ) R^{\alpha -n} r^{1-\alpha } R^{n-1} \ge c(n,\gamma ,\alpha ) \left( \frac{r}{R} \right)^{1-\alpha },\) 
and the last term tends to infinity as \(r \rightarrow 0.\)
First we show that (REF ) holds for every \(\alpha \)  with \(0<\alpha <\min \lbrace 1, \frac{n\gamma }{1+\gamma } \rbrace .\) 
This implies that
\(\operatorname{{Mu}}(E) \ge \min \lbrace 1, \frac{n \gamma }{1+\gamma } \rbrace >0\) ,
and thus \(E\)  is weakly porous, by Corollary REF .
Fix \(0<\alpha <\min \lbrace 1, \frac{n\gamma }{1+\gamma } \rbrace \)  and let
\(B=B(x,R) \subset {\mathbb {R}}^n\)  be a ball with \(x\in E,\)  and let \(0<r<h_E(B)\) .
We suppose first that \(B\)  is contained in \({\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,1).\( Let \) k\( be the largest number in \) N\( and \) N\( be the smallest number in \) N\( such that \) B \(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B ( 0, k- ) B(0, N- ).\( We interpret \) N-=0\( and \) B(0, N- ) =\( when \) 0 \(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B.\( It is clear that \) N k+2\(, since the center \) x\( of \) B\( belongs to \) E\(. In the case\) N= k+2\( we have \) xS(k+1)-\(,and (\ref {e.muckdim}) follows immediately from Lemma~\ref {lemmaonecircle}.Hence we may assume that \) N k+3.\( Also, observe that\begin{equation}h_E(B) \le \tfrac{1}{2}\left( k^{-\gamma }-(k+1)^{-\gamma } \right) \le \tfrac{\gamma }{2} k^{-\gamma -1}\end{equation}and\begin{equation}R \ge \tfrac{1}{2} \left( (k+1)^{-\gamma }-(N-1)^{-\gamma } \right)\ge \tfrac{\gamma }{2}(N-k-2)(N-1)^{-\gamma -1}.\end{equation}Now we study two cases.\)
(i) Suppose \(\operatorname{dist}(\lbrace 0 \rbrace , B ) > \operatorname{diam}(B).\)  We have the estimates
\((k+1)^{-\gamma } \le \sup _{x\in B} \vert x\vert \le \operatorname{dist}(\lbrace 0 \rbrace , B ) + \operatorname{diam}(B)\le 2 \operatorname{dist}(\lbrace 0 \rbrace , B ) \le 2 (N-1)^{-\gamma } ,\) 
and so \(N-1\le C(\gamma ) (k+1).\)  Then we have
\(\vert E_r \cap B \vert & \le \sum _{j=k}^N \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \le C(n) \sum _{j=k}^N r R^{n-1} \le C(n) (N-k+1) r R^{n-1}.\) 

The previous observation, together with
() and (), leads us to
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } & \le C(n,\gamma ) k^{-(1+\gamma ) \alpha } \, r^{1-\alpha } R^{-1}(N-k+1) \\&\le C(n,\gamma ) k^{-(1+\gamma ) \alpha } \, r^{1-\alpha }(N-1)^{1+\gamma }\\&\le C(n,\gamma ) k^{-(1+\gamma ) \alpha } \, r^{1-\alpha }(k+1)^{1+\gamma }\\& \le C(n,\gamma ) \left(rk^{1+\gamma }\right)^{1-\alpha }.\) 

The last term is bounded by a constant \(C(n, \gamma ,\alpha )\)  because \(\alpha \le 1\)  and \(r \le C(\gamma ) k^{-1-\gamma }.\)
(ii) Now suppose \(\operatorname{dist}(\lbrace 0 \rbrace , B ) \le \operatorname{diam}(B).\)  Then we have
\((2k)^{-\gamma } \le (k+1)^{-\gamma } \le \operatorname{dist}(\lbrace 0 \rbrace , B ) + \operatorname{diam}(B) \le 2 \operatorname{diam}(B),\) 
and hence \(k^{-\gamma } \le 2^{1+\gamma } \operatorname{diam}(B).\)  Given \(0<r<h_E(B),\)  denote by \(j_0\in {\mathbb {N}}\)  the
smallest number for which
\(2r \ge j_0^{-\gamma }-\left( j_0+1 \right)^{-\gamma } \ge C(\gamma )(j_0+1)^{-\gamma -1}.\) 
Notice that \(k< j_0\)  and, by the definition of \(j_0,\)  we also have
\(r \le (j_0-1)^{-\gamma }- j_0^{-\gamma }\le C(\gamma ) (j_0-1)^{-\gamma -1}\le C(\gamma )j_0^{-\gamma -1}.\) 
This observation permits us to write
\(\vert E_r \cap B \vert & \le \vert B \cap A_{k^{-\gamma } -r}^{k^{-\gamma }} \vert + \vert B(0,j_0^{-\gamma }+r) \cap B \vert + \sum _{j=k+1}^{j_0-1} \vert A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \cap B \vert \\& \le C(n) \biggl ( \left( j_0^{-\gamma }+r \right)^n + \sum _{j=k }^{j_0-1} r \left( j^{-\gamma }+r \right)^{n-1} \biggr ) .\) 

Using the inequalities \(0<\alpha <\min \lbrace 1, \frac{n\gamma }{1+\gamma } \rbrace ,\)  \(k^{-\gamma } \le C(\gamma ) R\) , \(c(\gamma )j_0^{-1-\gamma }\le r \le C(\gamma )j_0^{-1-\gamma },\)  and \( h_E(B) \le C(\gamma ) k^{-1-\gamma },\)  we obtain
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } & \le C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } r^{-\alpha }\biggl ( \left( j_0^{-\gamma }+r \right)^n + \sum _{j=k }^{j_0-1} r \left( j^{-\gamma }+r \right)^{n-1} \biggr ) \\& \le C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } r^{-\alpha } \biggl ( j_0^{-n\gamma } + \sum _{j=k }^{j_0-1} r \left( j^{-\gamma }+r \right)^{n-1} \biggr ) \\& \le C(n,\gamma ) \biggl ( \left( k j_0^{-1} \right)^{n\gamma -(1+\gamma ) \alpha } + k^{n \gamma -(1+\gamma )\alpha } \sum _{j=k }^{j_0-1} r^{1-\alpha } \left( j^{-\gamma }+r \right)^{n-1} \biggr ) \\& \le C(n,\gamma )+ C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } \sum _{j=k }^{j_0-1} j^{-(1-\alpha )(1+\gamma )} \left( j^{-\gamma }+j^{-1-\gamma }\right)^{n-1} \\& \le C(n,\gamma )+ C(n,\gamma ) k^{n \gamma -(1+\gamma )\alpha } \sum _{j=k }^{\infty } j^{-1-n\gamma +(1+\gamma ) \alpha } \le C(n,\gamma ,\alpha ),\) 

where the last inequality follows
by comparing the series to
\(\int _{k}^\infty t^{-1-n\gamma +(1+\gamma ) \alpha }\,dt\) , bearing in mind that \(\alpha < \frac{n\gamma }{1+\gamma }.\) 
The cases (i) and (ii) together show that
(REF ) holds when \(B\subset {\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,1)\(.\)
Now suppose that \(B=B(x,R)\)  is not contained in \({\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,1).\(In the case \) r1-2-2\( we use the fact that \) n-> 0\( to estimate{\begin{@align*}{1}{-1}\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert }& \le C(n)\vert E_r \vert r^{-\alpha } R^{\alpha -n}\le C(n)\vert {\mathchoice{\unknown. {\displaystyle B}\hspace{0.0pt}\vrule height1.593pt}{\hspace{0.0pt}}{0}{.}0\overline{\phantom{\displaystyle B}\displaystyle B}\hspace{0.0pt}}\end{@align*}{B}}{\unknown. {\textstyle B}\hspace{0.0pt}\vrule height1.593pt}\hspace{0.0pt}\overline{\phantom{\textstyle B}\textstyle B}\hspace{0.0pt}\)\(\) B\(\) B\(\) 

B
\(\)B\(\) B\(\) 

B
\(\)B(0,r+1)r- R-n
C(n,)(rR)n-C(n,,).

In the sequel, we will assume that
\(r< \frac{1-2^{-\gamma }}{2}\) .
If \(x\in E \setminus S^1,\)  then \(R\ge h_E(B) \ge c(\gamma ) R \ge c(\gamma )\) 
and
\(\vert E_r \cap B \vert \le \vert E_r \cap B(0,1)\vert + \vert E_r \setminus B(0,1)\vert \le \vert E_r \cap B(0,1)\vert +C(n)r.\) 
Therefore
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert }\le C(n)(r+\vert E_r \cap B(0,1)\vert ) r^{-\alpha } R^{\alpha -n}\le C(n,\gamma ,\alpha ) R^{\alpha -n} \le C(n,\gamma ,\alpha ),\) 
where the second inequality follows
by using the above case (ii) with \(B=B(0,1)\) .
If \(x\in S^1\)  and \(R\ge \frac{1-2^{-\gamma }}{2},\)  then we can repeat the preceding argument to show that (REF ) holds, and finally, if \(x\in S^1\)  and \(R<\frac{1-2^{-\gamma }}{2},\) 
then (REF ) holds by Lemma REF .
Next we show that \(\operatorname{{Mu}}(E) \le \min \lbrace 1, \frac{n \gamma }{1+\gamma } \rbrace .\) 
The bound \(\operatorname{{Mu}}(E) \le 1\)  follows from Lemma REF .
Let \(\alpha > \frac{n \gamma }{1+\gamma }\)  and consider the ball \(B=B(0,1).\) 
Then \(h_E(B)=\frac{1-2^{-\gamma }}{2}=C(\gamma ).\)  Given \(0<r<\frac{1}{100}\) , let \(j_0\in {\mathbb {N}}\)  be the
smallest number for which \(2r \ge j_0^{-\gamma }-(j_0+1)^{-\gamma }.\)  Then \(r\)  is comparable to \(c(\gamma )j_0^{-1-\gamma }\)  and the annuli \(\lbrace A_{j^{-\gamma }-r}^{j^{-\gamma }+r} \rbrace _{j=1}^{j_0}\)  are pairwise disjoint.
For sufficiently small \(r,\)  we thus have
\(\left( \frac{h_E(B)}{r} \right)^{\alpha } \frac{\vert E_r \cap B \vert }{\vert B \vert } &\ge c(n,\gamma ,\alpha ) r^{-\alpha }\sum _{j=2}^{j_0-1} \left( \left( j^{-\gamma }+r \right)^{n}- \left( j^{-\gamma }-r \right)^{n} \right) \\& \ge c(n,\gamma ,\alpha ) r^{1-\alpha } \sum _{j=2}^{j_0-1} \left( j^{-\gamma }-r \right)^{n-1} \\& \ge c(n,\gamma ,\alpha ) r^{1-\alpha } j_0 \left( (j_0-1)^{-\gamma }-r \right)^{n-1} \\& \ge c(n,\gamma ,\alpha ) r^{1-\alpha } j_0 \left( j_0^{-\gamma }-j_0^{-\gamma -1} \right)^{n-1} \ge c(n,\gamma ,\alpha ) r^{1-\alpha } j_0^{1-\gamma (n-1)} \\&\ge c(n,\gamma ,\alpha ) j_0^{(1-\alpha )(-1-\gamma )} j_0^{1-\gamma (n-1)} = c(n,\gamma ,\alpha ) j_0^{(1+\gamma ) \alpha -n \gamma }.\) 

The last term goes to infinity as \(r\rightarrow 0,\)  since \(\alpha > \frac{n \gamma }{1+\gamma }\) .
Hence (REF ) does not
hold if \(\alpha > \frac{n \gamma }{1+\gamma }\) , showing
that \(\operatorname{{Mu}}(E) \le \frac{n \gamma }{1+\gamma }\) .

\(A_p\) -distance set that is not weakly porous
In this section we construct a set \(E \subset {\mathbb {R}}\)  such
that \(\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p\setminus A_1\) 
for all \(0<\alpha <1\)  and all \(1<p<\infty \) ; see
Theorem REF .
Recall that we abbreviate \(d_E=\operatorname{dist}(\cdot ,E)\) .
Let \(E_0=\lbrace 0,1 \rbrace \)  and write \(t_n = 1-\frac{1}{2n}\)  for every \(n \in {\mathbb {N}}\) .
Then, for every \(n\in {\mathbb {N}},\) 
the set \(E_n\)  is defined as \(E_n=E_{n-1} \cup E_{n-1}^1 \cup E_{n-1}^2,\)  where:

\(E_{n-1}^1\)  is a translation of \(E_{n-1}\)  dilated by the factor \(t_n\)  and whose first point is the last point of \(E_{n-1},\)

\(E_{n-1}^2\)  is a translation of \(E_{n-1}\)  whose first point is the last point of \(E_{n-1}^1.\)

Finally, we define \(E^+= \bigcup _{n=0}^\infty E_n\)  and \(E= E^+ \cup \left( - E^+ \right).\)  Here \(-E^+\)  is the reflection of \(E^+\)  with respect to the origin. We let \(Q_n,\)  \(Q_n^1,\)  and \(Q_n^2\)  denote the smallest intervals containing \(E_n\) , \(E_n^1,\)  and \(E_n^2\)  respectively, for every \(n\in {\mathbb {N}}\cup \lbrace 0 \rbrace .\) 
See Figure REF  for an illustration of the first steps of the construction.
<FIGURE>During the rest of this section, we prove the following theorem
for the set \(E\) .
Theorem 8.1 
Let \(E\subset {\mathbb {R}}\)  be as constructed above.
Then it holds for all \(0<\alpha <1\)  and all \(1<p<\infty \) 
that \(\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p \setminus A_1\) .
In particular,
the set \(E\)  is not weakly porous and \(\operatorname{{Mu}}(E)=0\) .

Let \(0<\alpha <1\)  and \(1<p<\infty \) . We show in Lemma REF  that
\(\operatorname{dist}(\cdot ,E)^{-\alpha } \notin A_1\) , and the claim
\(\operatorname{dist}(\cdot ,E)^{-\alpha } \in A_p\) 
follows from Lemma REF .
Since \(\operatorname{dist}(\cdot ,E)^{-\alpha } \notin A_1\)  for every \(\alpha >0\) , the set
\(E\)  is not weakly porous by Theorem REF ,
and thus Corollary REF  implies
that \(\operatorname{{Mu}}(E)=0\) .
We say that a closed interval \(I\)  is an edge of \(E\) 
if the endpoints of \(I\)  are two consecutive points of \(E.\) 
For every \(n\in {\mathbb {N}}\cup \lbrace 0 \rbrace ,\)  the following
properties hold:

Each of the cubes \(Q_n,\)  \(Q_n^1,\)  and \(Q_n^2\)  has \(3^n\)  edges of \(E,\)  of which the middle ones for \(n\ge 1\)  have lengths equal to \(t_1 t_2 \cdots t_n,\)  \(t_1 t_2 \cdots t_n t_{n+1},\)  and \(t_1 t_2 \cdots t_n\) , respectively.

Each of the cubes \(Q_n\)  and \(Q_n^2\)  contains translated copies of the intervals \(Q_0, \ldots , Q_{n-1}\)  distributed in a palindromic manner: both \(Q_n\)  and \(Q_n^2\)  contain from left to right as well as from right to left intervals \(Q_0^* \subset Q_1^* \subset \cdots \subset Q_{n-1}^*\)  that are translated copies of \(Q_0 \subset Q_1 \subset \cdots \subset Q_{n-1}\) , respectively.

Each cube \(Q_n^1\)  contains from left to right as well as from right to left intervals \(t_N Q_0^* \subset t_N Q_1^* \subset \cdots \subset t_N Q_{n-1}^*\)  that are translated copies of \(Q_0 \subset Q_1 \subset \cdots \subset Q_{n-1}\)  dilated by \(t_N.\)

\(d_E= d_{E_n}\)  on \(Q_n.\)

\(\vert Q_n \vert = (2+t_n) \vert Q_{n-1} \vert \)  for every \(n\in {\mathbb {N}}.\)

Lemma 8.2 
For every \(n \in {\mathbb {N}}\)  and every \(\beta >-1,\)  we have
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E(x)^\beta \,dx = \frac{2+t_n^{1+\beta }}{2+t_n} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_{n-1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{n-1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{n-1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{n-1}}} d_E(x)^\beta \,dx.\) 

Let \(n\in {\mathbb {N}}\)  and \(\beta >-1\) .
By the construction of \(E\)  and the definition of \(Q_n\) , we obtain
\(\int _{Q_n} d_E^\beta & =\int _{Q_n} d_{E_n}^\beta = \int _{Q_{n-1}} d_{E_{n-1}}^\beta + \int _{Q_{n-1}^1} d_{E_{n-1}^1}^\beta + \int _{Q_{n-1}^2} d_{E_{n-1}^2}^\beta \\& = \left( 2 + t_n^{1+\beta } \right) \int _{Q_{n-1}} d_{E_{n-1}}^\beta = \left( 2 + t_n^{1+\beta } \right) \int _{Q_{n-1}} d_{E}^\beta .\) 

The claim follows by combining the above identity with the fact \(\vert Q_n \vert = (2+t_n) \vert Q_{n-1} \vert \) .
Lemma 8.3 
For every \(0<\alpha <1\)  and \(1<p<\infty \) , there exists \(N_0\in {\mathbb {N}}\) , only depending on \(\alpha \)  and \(p\) , for which
\(\log \left( \frac{ 2+t_n^{1-\alpha }}{2+t_n} \right) \ge \frac{\alpha }{ 12n} \quad \text{and} \quad \log \left[\left( \frac{ 2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1}\right] \le \frac{\alpha ^2 p}{18(p-1) n^2}\) 
for every \(n \ge N_0.\)

Consider the functions
\(f(t)=\log \left( \frac{ 2+t^{1-\alpha }}{2+t} \right),\qquad g(t)=\log \left[\left( \frac{ 2+t^{1-\alpha }}{2+t} \right) \left( \frac{2+t^{1+\frac{\alpha }{p-1}}}{2+t} \right)^{p-1}\right]\) 
for \(t>0.\)  These functions satisfy \(f(1)=0,\)  \(f^{\prime }(1)= -\frac{\alpha }{3},\)  \(g(1)=g^{\prime }(1)=0\)  and \(g^{\prime \prime }(1)= \frac{2\alpha ^2 p}{9(p-1)}.\)  Let \(\varepsilon \in (0,1/2)\)  be small enough so that \(\vert t-1 \vert \le \varepsilon \)  implies
\(\vert f(t)-f(1)-f^{\prime }(1)(t-1) \vert \le \frac{\alpha }{6} \vert t-1 \vert \) 
and
\(\vert g(t)-g(1)-g^{\prime }(1)(t-1)-\tfrac{1}{2}g^{\prime \prime }(1)(t-1)^2 \vert \le \frac{\alpha ^2 p}{9(p-1)} \vert t-1 \vert ^2.\) 
Taking \(N_0 \in {\mathbb {N}}\)  large enough so that \(N_0 \ge 1/(2\varepsilon )\) 
it follows that \(\vert 1-t_n\vert \le \varepsilon \)  for every \(n \ge N_0,\) 
and so the above estimates yield
\(f(t_n) \ge \frac{\alpha }{12n} \qquad \text{and} \qquad g(t_n)\le \frac{\alpha ^2 p}{18(p-1) n^2}. \) 
Lemma 8.4 
For every \(0<\alpha <1\) , the weight \(d_E^{-\alpha }\)  does not belong to \(A_1.\)

Let \(N_0\)  be the constant in Lemma REF 
with, say, \(p=2\) ; the value of \(p\)  is irrelevant here. Applying repeatedly
Lemma REF , we obtain, for every \(n\in {\mathbb {N}},\)
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E^{-\alpha } = \left( \prod _{k=1}^n \frac{2+t_k^{1-\alpha }}{2+t_k} \right) \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha } \ge \left( \prod _{k=N_0}^n \frac{2+t_k^{1-\alpha }}{2+t_k} \right) \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }.\) 
By the first inequality of Lemma REF , we have
\(\log \left( \prod _{k=N_0}^n \frac{2+t_k^{1-\alpha }}{2+t_k} \right) = \sum _{k=N_0}^n \log \left( \frac{2+t_k^{1-\alpha }}{2+t_k} \right) \ge \sum _{k=N_0}^n \frac{\alpha }{ 12 k},\) 
and it follows that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E^{-\alpha }\ge \exp \left(\sum _{k=N_0}^n \frac{\alpha }{ 12 k}\right)\,\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }.\) 
Since the harmonic series
diverges, we see that
\(\lim _{n\rightarrow \infty } \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_n}} d_E^{-\alpha } =\infty \) . On the other hand, each \(Q_n\)  contains edges of \(E\)  of length equal to 1, and thus \(\operatornamewithlimits{ess\, inf}_{Q_n} d_E^{-\alpha } =2^\alpha .\)  We conclude that \(d_E^{-\alpha } \notin A_1.\)
Lemma 8.5 
For every \(0<\alpha <1\)  and \(1<p<\infty \) ,
there exists a constant \(\widehat{C} = \widehat{C}(\alpha ,p)>0\)  such that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E(x)^{-\alpha }\,dx \left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E(x)^{\frac{\alpha }{p-1}}\,dx \right)^{p-1} \le \widehat{C}\) 
for every \(N\in {\mathbb {N}}\cup \lbrace 0 \rbrace \) .

For \(N=0\)  the claim is clear. Assume that \(N\ge 1\) . By Lemma REF ,
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} & = \left( \prod _{n=1}^{N} \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \prod _{n=1}^{N} \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \\& =\prod _{n=1}^{N} \left( \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1}\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1}.\) 

Let \(N_0=N_0(\alpha ,p)\in {\mathbb {N}}\)  be as in Lemma REF .
Then
\(\log & \prod _{n=1}^{N} \left( \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1} \\& \le \sum _{n=1}^{N_0-1} \log \left[\left( \frac{2+t_n^{1-\alpha }}{2+t_n} \right) \left( \frac{2+t_n^{1+\frac{\alpha }{p-1}}}{2+t_n} \right)^{p-1}\right]+ \sum _{n=N_0}^N \frac{\alpha ^2 p}{18(p-1) n^2},\) 

where the right-hand side is bounded from above by a constant \(C_1 = C_1(\alpha ,p)\)  independent of \(N\) .
Hence,
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le e^{C_1} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_0}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1},\) 
and the claim follows.
Lemma 8.6 
For every \(0<\alpha <1\)  and \(1<p<\infty \) ,
there exists a constant \(C = C(\alpha ,p)>0\)  such that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{-\alpha }\,dx \left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{\frac{\alpha }{p-1}}\,dx \right)^{p-1} \le C\) 
for every cube \(Q \subset [0, +\infty ).\)

Observe that \(Q\subset Q_N\)  for some \(N\in {\mathbb {N}}\) . When \(Q\)  contains at most 4 points of \(E,\)  it is straightforward to see that the distance \(d_E\)  satisfies (REF ) for \(Q \)  and with some constant \(C_1\)  only depending on \(\alpha \)  and \(p.\)  This includes the case where \(Q \)  is contained in \(Q_1.\)
We prove by induction on \(N\)  that \(d_E\)  satisfies (REF ) for every interval \(Q \subset Q_N\)  with the constant \(C=\max \lbrace 12^p \widehat{C}, C_1\rbrace \) , where \(\widehat{C}\)  is the constant in Lemma REF . The case \(N=1\)  has already been proved since \(C\ge C_1\) . Hence, we assume that the claim holds for all \(n=1, \ldots , N-1,\) 
and we need to verify the claim for all intervals \(Q\)  contained in \(Q_N.\)
The case where \(Q \subset Q_{N-1}\)  follows from the induction hypothesis. Thus we may and do assume that \(Q\)  is not contained in \(Q_{N-1}.\)  We do a case study.
(i): \(Q\)  is contained in one of the intervals \(Q_{N-1}^1\) , \(Q_{N-1}^2.\)  In the first case, the cube \(Q\subset Q_{N-1}^1\)  can be written as \(Q=t_N Q^*\) , where \(Q^*\)  is a translation of an interval \(\widehat{Q}\)  contained in \(Q_{N-1}.\)  Then \(\vert Q \vert = t_N \vert \widehat{Q} \vert \)  and \(\int _Q d_E^\beta = t_N^{1+\beta } \int _{\widehat{Q}} d_E^\beta \)  for every \(\beta >-1.\)  This gives
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} = \left( t_N^{-\alpha } \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{-\alpha } \right) \left( t_N^{\frac{\alpha }{p-1}} \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} =\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{\widehat{Q}}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le C,\) 
where the last inequality holds by the induction hypothesis. In the second case we have \(Q\subset Q_{N-1}^2,\)  and inequality (REF ) follows from the induction hypothesis since \(Q\)  is now translation of an interval \(\widehat{Q}\)  contained in \(Q_{N-1}\) .
(ii): \(Q\)  intersects both \(Q_{N-1}\)  and \(Q_{N-1}^2.\)  This implies that \(Q\)  contains \(Q_{N-1}^1,\)  and so
\(\vert Q \vert \ge |Q_{N-1}^1|= t_N \vert Q_{N-1} \vert = \frac{t_N}{2+t_N} \vert Q_N \vert \ge \frac{1}{6} \vert Q_N \vert .\) 
Using this estimate together with Lemma REF , we obtain
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} &\le \left( \frac{6}{ \vert Q_N \vert }\int _{Q } d_E^{-\alpha } \right)\left( \frac{6}{\vert Q_N \vert }\int _{Q } d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \\&\le 6^p \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_N}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le 6^p \widehat{C}.\) 
(iii): \(Q\)  contains one of the intervals \(Q_{N-1},\)  \(Q_{N-1}^1\) , \(Q_{N-1}^2.\)  In this case
\(|Q| \ge t_N \vert Q_{N-1} \vert \ge \frac{1}{6} \vert Q_N \vert .\)  Using that \(Q \subset Q_N,\)  the desired estimate follows as in the case (ii).
(iv): Assume that \(Q \cap Q_{N-1} \ne \emptyset \ne Q \cap Q_{N-1}^1\)  but \(Q \cap Q_{N-1}^2=\emptyset .\)  By the construction of \(Q_{N-1}\) , we can find \(m\in \lbrace -1, 0, \ldots , N-2 \rbrace \)  so that \( Q_m^* \subset Q \cap Q_{N-1} \subset Q_{m+1}^* ,\)  where \(Q_m^*\)  and \(Q_{m+1}^*\)  are translations of \(Q_m\)  and \(Q_{m+1}\)  respectively, and we use the notation \(Q_{-1}^*=\emptyset .\)  This implies \(\vert Q \cap Q_{N-1} \vert \ge \vert Q_{m} \vert .\) 
Similarly, by the construction of \(Q_{N-1}^1,\)  there exists \(n \in \lbrace -1, 0, \ldots , N-2 \rbrace \)  so that \(t_N Q_n^* \subset Q \cap Q_{N-1}^1 \subset t_N Q_{n+1}^* ,\)  where \(Q_n^*\)  and \(Q_{n+1}^*\)  are translations of \(Q_n\)  and \(Q_{n+1}\) , respectively, and so \(\vert Q \cap Q_{N-1}^1 \vert \ge t_N \vert Q_n \vert .\)  Now define \(M=\max \lbrace m,n\rbrace .\)  If \(M=-1,\)  then \(Q\)  intersects at most 2 edges of \(E,\)  and the desired estimate follows with the constant \(C_1\)  from the beginning of the proof. If \(M \ge 0,\)  then we have \(Q \cap Q_{N-1} \subset Q_{M+1}^*\)  and \(Q \cap Q_{N-1}^1 \subset t_N Q_{M+1}^*,\)  and so
\(\int _Q d_E^\beta \le \int _{Q_{M+1} }d_E^\beta + t_N^{1+\beta }\int _{Q_{M+1} }d_E^\beta = \left( 1+ t_N^{1+\beta } \right) \int _{Q_{M+1} }d_E^\beta \le 2 \int _{Q_{M+1} }d_E^\beta ,\) 
for every \(\beta >-1.\)  On the other hand,
\(\vert Q \vert &= \vert Q \cap Q_{N-1} \vert + \vert Q \cap Q_{N-1}^1 \vert \ge \vert Q_{m} \vert + t_N \vert Q_{n} \vert \\& \ge t_N \vert Q_{M} \vert = \frac{t_N}{2+ t_{M+1}} \vert Q_{M+1} \vert \ge \frac{|Q_{M+1}|}{6}.\) 

This leads us to
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1}\le 12^p \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}} d_E^{-\alpha }\biggl ( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q_{M+1}}} d_E^{\frac{\alpha }{p-1}} \biggr )^{p-1} \le 12^p \widehat{C},\) 
where the last inequality follows from Lemma REF .
(v): Assume that \(Q \cap Q_{N-1}^1 \ne \emptyset \ne Q \cap Q_{N-1}^2\)  but \(Q \cap Q_{N-1}=\emptyset .\)  Recall that \(Q_{N-1}^2\)  is a translation of \(Q_{N-1}\)  that contains, from left to right, translated copies \(Q_0^* \subset Q_1^* \subset \cdots \subset Q_{N-2}^* \subset Q_{N-1}^*\)  of \(Q_0\subset Q_1 \subset \cdots \subset Q_{N-2} \subset Q_{N-1}\) , respectively. In addition, \(Q_{N-1}^1\)  contains, from right to left, translated copies \(t_N Q_{N-1}^* \supset t_N Q_{N-2}^* \supset \cdots \supset t_N Q_1^* \supset t_N Q_0^* \)  of \(Q_{N-1} \supset Q_{N-2} \supset \cdots \supset Q_1 \supset Q_0\)  dilated by \(t_N.\)  Now, the argument is identical to the case (iv).
Lemma 8.7 
Let \(0<\alpha <1\)  and \(1<p<\infty \) , and let \(C=C(\alpha ,p)\)  be the constant in Lemma REF .
Then
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{-\alpha }\,dx \left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E(x)^{\frac{\alpha }{p-1}}\,dx \right)^{p-1} \le 2^p C\) 
for every cube \(Q \subset {\mathbb {R}}\) , and so \(d_E^{-\alpha } \in A_p.\)

Given a cube \(Q \subset {\mathbb {R}}\) , we write \(Q^+=Q \cap [0,+ \infty )\)  and \(Q^-=Q \cap (-\infty , 0].\) 
Let \(Q^*\)  be the largest of the cubes \(Q^+\)  and \(-Q^-\) , that is, \(Q^*\in \lbrace Q^+, -Q^- \rbrace \) 
and \(Q^+ \cup -Q^-\subset Q^*\) . Here
\(-Q^{-}\)  denotes the reflection of \(Q^{-}\)  with respect to the origin.
Because \(E\)  is symmetric with respect to the origin, we can write
\(\int _Q d_E^{-\alpha } = \int _{Q^+} d_E^{-\alpha } + \int _{Q^-} d_E^{-\alpha } = \int _{Q^+} d_E^{-\alpha } + \int _{-Q^-} d_E^{-\alpha } \le 2 \int _{Q^*} d_E^{-\alpha }.\) 
The same argument shows that \(\int _Q d_E^{\frac{\alpha }{p-1}} \le 2 \int _{Q^*} d_E^{\frac{\alpha }{p-1}}.\)  Because \(\vert Q\vert \ge \vert Q^*\vert \)  and \(Q^*\)  is contained in \([0,\infty ),\)  we can use Lemma REF  to conclude
that
\(\mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q }} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le 2^p \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}} d_E^{-\alpha }\left( \mathchoice{\mathop {\hspace{1.99997pt}\vrule width 0.6em height 0.69678ex depth -0.58065ex\hspace{-8.00003pt}}\nolimits _{\hspace{-3.99994pt}Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678exdepth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}}{\mathop {\hspace{1.00006pt}\vrule width 0.5em height 0.69678ex depth -0.60387ex\hspace{-6.00006pt}}\nolimits _{Q^*}} d_E^{\frac{\alpha }{p-1}} \right)^{p-1} \le 2^p C. \) 
